Weighted norm inequalities for integral transforms with kernels - PowerPoint PPT Presentation

Weighted norm inequalities for integral transforms with kernels bounded by power functions Alberto Debernardi Centre de Recerca Matem` atica, Barcelona 6th Workshop on Fourier Analysis and Related Topics University of P´ ecs, 26 August 2017 A. Debernardi–CRM WNI’s for integral transforms 1 / 22

Brief history: weighted norm inequalities for the Fourier transform For the Fourier transform � f ( x ) e − ixy dx, � f ( y ) = R it was proved in the 1980s (Muckenhoupt, Jurkat-Sampson) that the weighted norm inequality � � � 1 /q � � � 1 /p f ( y ) | q dy v ( x ) | f ( x ) | p dx � � u ( y ) | � f � q,u := ≤ C =: � f � p,v R R holds for every f with 1 < p ≤ q < ∞ and C > 0 independent of f provided that there exists D > 0 such that for every r > 0 � � 1 /r � 1 /q � � r � 1 /p ′ (1 /v ) ∗ ( x ) 1 − p ′ dx u ∗ ( y ) dy ≤ D. 0 0 A. Debernardi–CRM WNI’s for integral transforms 2 / 22

Problem Given an integral transform � ∞ Ff ( y ) = y c 0 x b 0 f ( x ) K ( x, y ) dx, y > 0 , b 0 , c 0 ∈ R , 0 where � ( xy ) b 1 , ( xy ) b 2 � | K ( x, y ) | � min , b 1 > b 2 . � 1 � ∞ 0 x b 0 + b 1 | f ( x ) | dx + 1 x b 0 + b 2 | f ( x ) | dx < ∞ . We also assume We want to give sufficient (and necessary, when possible) conditions for the weighted norm inequality � y − β Ff � q ≤ C � x γ f � p , 1 < p ≤ q < ∞ , (1) to hold, using an approach that does not involve decreasing rearrangements. A. Debernardi–CRM WNI’s for integral transforms 3 / 22

Examples 1. The Fourier transform is not a good example, since | K ( x, y ) | = | e 2 πixy | = 1 does not satisfy � ( xy ) b 1 , ( xy ) b 2 � | K ( x, y ) | � min , b 1 > b 2 . The cosine transform is also a bad example. 2. The sine transform satisfies � � | K ( x, y ) | = | sin xy | � min xy, 1 , i.e., b 1 = 1 > 0 = b 2 . A. Debernardi–CRM WNI’s for integral transforms 4 / 22

Examples: The Hankel transform 3. The classical Hankel transform of order α ≥ − 1 / 2 is defined as � ∞ x 2 α +1 f ( x ) j α ( xy ) dx, H α f ( y ) = 0 where j α is the normalized Bessel function of order α , represented through the series � ∞ ( − 1) n ( z/ 2) 2 n j α ( z ) = Γ( α + 1) n !Γ( n + α + 1) . n =0 The function j α satisfies the estimate � 1 , ( xy ) − α − 1 / 2 } . | j α ( xy ) | � min A. Debernardi–CRM WNI’s for integral transforms 5 / 22

Examples: The H α transform 4. The so-called H α transform is defined as � ∞ ( xy ) 1 / 2 f ( x ) H α ( xy ) dx, H α f ( y ) = α > − 1 / 2 , 0 where H α is the Struve function , defined as � z � α +1 ∞ � ( − 1) n ( z/ 2) 2 n H α ( z ) = Γ( n + 3 / 2)Γ( n + α + 3 / 2) . 2 n =0 A. Debernardi–CRM WNI’s for integral transforms 6 / 22

Examples: The H α transform 4. The so-called H α transform is defined as � ∞ ( xy ) 1 / 2 f ( x ) H α ( xy ) dx, H α f ( y ) = α > − 1 / 2 , 0 where H α is the Struve function , defined as � z � α +1 ∞ � ( − 1) n ( z/ 2) 2 n H α ( z ) = Γ( n + 3 / 2)Γ( n + α + 3 / 2) . 2 n =0 The Struve function satisfies the estimate � min { x α +1 , x − 1 / 2 } , α < 1 / 2 , | H α ( x ) | � min { x α +1 , x α − 1 } , α ≥ 1 / 2 , ◮ P. G. Rooney, Canad. J. Math (1980). A. Debernardi–CRM WNI’s for integral transforms 6 / 22

Known results Cosine transform (and Fourier transform): if Ff = � f or Ff = � f cos , then (1) holds if and only if β = γ + 1 /q − 1 /p ′ and max { 1 /q − 1 /p ′ , 0 } ≤ β < 1 /q. ◮ W. B. Jurkat–G. Sampson, Indiana Univ. Math. J. (1984); B. Muckenhoupt, Proc. Amer. Math. Soc. (1983). ◮ H. P. Heinig, Indiana Univ. Math. J. (1984). A. Debernardi–CRM WNI’s for integral transforms 7 / 22

Known results Cosine transform (and Fourier transform): if Ff = � f or Ff = � f cos , then (1) holds if and only if β = γ + 1 /q − 1 /p ′ and max { 1 /q − 1 /p ′ , 0 } ≤ β < 1 /q. ◮ W. B. Jurkat–G. Sampson, Indiana Univ. Math. J. (1984); B. Muckenhoupt, Proc. Amer. Math. Soc. (1983). ◮ H. P. Heinig, Indiana Univ. Math. J. (1984). Sine transform: if Ff = � f sin , (1) holds if and only if β = γ + 1 /q − 1 /p ′ and max { 1 /q − 1 /p ′ , 0 } ≤ β < 1 + 1 /q. ◮ D. Gorbachev, E. Liflyand, S. Tikhonov, Indiana Univ. Math. J. (to appear). A. Debernardi–CRM WNI’s for integral transforms 7 / 22

Known results Hankel transform: if Ff = H α f ( α ≥ − 1 / 2), then (1) holds if and only if β = γ − 2 α − 1 + 1 /q − 1 /p ′ and max { 1 /q − 1 /p ′ , 0 } − α − 1 / 2 ≤ β < 1 /q. ◮ P. Heywood, P. G. Rooney, Proc. Roy. Soc. Edinburgh (1984). ◮ L. de Carli, J. Math. Anal. Appl. (2008). A. Debernardi–CRM WNI’s for integral transforms 8 / 22

Known results H α transform: if Ff = H α f ( α > − 1 / 2), (1) holds provided that β = γ + 1 /q − 1 /p ′ and – for − 1 / 2 < α < 1 / 2, β ≥ max { 1 /q − 1 /p ′ , 0 } and 1 /q + α − 1 / 2 < β < 1 /q + α + 3 / 2; – for α ≥ 1 / 2, 1 /q + α − 1 / 2 < β < 1 /q + α + 3 / 2 . ◮ P. G. Rooney, Canad. J. Math. (1980). A. Debernardi–CRM WNI’s for integral transforms 9 / 22

Main result (sufficient conditions) The following states sufficient conditions for inequality (1) to hold. Theorem Let 1 < p ≤ q < ∞ . If the integral transform � ∞ Ff ( y ) = y c 0 x b 0 f ( x ) K ( x, y ) dx, y > 0 , b 0 , c 0 ∈ R , 0 � ( xy ) b 1 , ( xy ) b 2 � satisfies | K ( x, y ) | � min , with b 1 > b 2 , then the inequality � y − β Ff � q ≤ C � x γ f � p , 1 < p ≤ q < ∞ , holds for every f , provided that β = γ + c 0 − b 0 + 1 q − 1 1 q + c 0 + b 2 < β < 1 p ′ , q + c 0 + b 1 . A. Debernardi–CRM WNI’s for integral transforms 10 / 22

Sharpness and necessity conditions � ( xy ) b 1 , ( xy ) b 2 � If instead of | K ( x, y ) | � min , there holds � ( xy ) b 1 , ( xy ) b 2 � K ( x, y ) ≍ min , the latter theorem can be improved. Theorem Let 1 < p ≤ q < ∞ . If the integral transform � ∞ Ff ( y ) = y c 0 x b 0 f ( x ) K ( x, y ) dx, y > 0 , b 0 , c 0 ∈ R , 0 � ( xy ) b 1 , ( xy ) b 2 � satisfies | K ( x, y ) | � min , with b 1 > b 2 , then the inequality � y − β Ff � q ≤ C � x γ f � p holds for every f with β = γ + c 0 − b 0 + 1 q − 1 1 q + c 0 + b 2 < β < 1 p ′ , q + c 0 + b 1 . A. Debernardi–CRM WNI’s for integral transforms 11 / 22

Sharpness and necessity conditions � ( xy ) b 1 , ( xy ) b 2 � If instead of | K ( x, y ) | � min , there holds � ( xy ) b 1 , ( xy ) b 2 � K ( x, y ) ≍ min , the latter theorem can be improved. Theorem Let 1 < p ≤ q < ∞ . If the integral transform � ∞ Ff ( y ) = y c 0 x b 0 f ( x ) K ( x, y ) dx, y > 0 , b 0 , c 0 ∈ R , 0 � ( xy ) b 1 , ( xy ) b 2 � satisfies K ( x, y ) ≍ min , with b 1 > b 2 , then the inequality � y − β Ff � q ≤ C � x γ f � p holds for every f if and only if β = γ + c 0 − b 0 + 1 q − 1 1 q + c 0 + b 2 < β < 1 p ′ , q + c 0 + b 1 . A. Debernardi–CRM WNI’s for integral transforms 11 / 22

Examples We can get the following sufficient conditions for (1): Cosine transform: no sufficient conditions! ( b 1 � > b 2 ). max { 1 /q − 1 /p ′ , 0 } ≤ β < 1 /q. A. Debernardi–CRM WNI’s for integral transforms 12 / 22

Examples We can get the following sufficient conditions for (1): Cosine transform: no sufficient conditions! ( b 1 � > b 2 ). max { 1 /q − 1 /p ′ , 0 } ≤ β < 1 /q. Sine transform: 1 /q < β < 1 + 1 /q max { 1 /q − 1 /p ′ , 0 } ≤ β < 1 + 1 /q. A. Debernardi–CRM WNI’s for integral transforms 12 / 22

Examples We can get the following sufficient conditions for (1): Cosine transform: no sufficient conditions! ( b 1 � > b 2 ). max { 1 /q − 1 /p ′ , 0 } ≤ β < 1 /q. Sine transform: 1 /q < β < 1 + 1 /q max { 1 /q − 1 /p ′ , 0 } ≤ β < 1 + 1 /q. Hankel transform of order α > − 1 / 2: 1 /q − α − 1 / 2 < β < 1 /q max { 1 /q − 1 /p ′ , 0 } − α − 1 / 2 ≤ β < 1 /q. A. Debernardi–CRM WNI’s for integral transforms 12 / 22

Examples H α transform ( α > − 1 / 2): 1 /q < β < 1 /q + α + 3 / 2 , − 1 / 2 < α < 1 / 2 , 1 /q + α − 1 / 2 < β < 1 /q + α + 3 / 2 , α ≥ 1 / 2 . Recall the known sufficient conditions: – For − 1 / 2 < α < 1 / 2, β ≥ max { 1 /q − 1 /p ′ , 0 } and 1 /q + α − 1 / 2 < β < 1 /q + α + 3 / 2 . – For α ≥ 1 / 2, 1 /q + α − 1 / 2 < β < 1 /q + α + 3 / 2 . A. Debernardi–CRM WNI’s for integral transforms 13 / 22

Transforms with kernel represented by power series Recall that � y − β � f � q ≤ C � x γ f � p (2) holds for every f if and only if β = γ + 1 /q − 1 /p ′ and max { 1 /q − 1 /p ′ , 0 } ≤ β < 1 /q. (3) � It was proved by Sadosky and Wheeden that if R f = 0, then (2) holds for 1 /q < β < 1 + 1 /q , additionally to (3). ◮ C. Sadosky and R. L. Wheeden, Trans. Amer. Mat. Soc. (1987). A. Debernardi–CRM WNI’s for integral transforms 14 / 22